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When solving a route using Network Analyst, the output route is a measured polyline, but all vertices have measure values of NaN, not a number.
There are non-zero impedance values on either the junctions or turns in the network. When a route contains these impedances, it is not possible to create a polyline with measure values indicative of the elapsed impedance along the length of the route.
Export the route to a feature class, and then manually edit the measure values along the route.